WASHINGTON—The White House said Thursday it could have better handled the domestic-abuse allegations that led to the resignation of staff secretary Rob Porter, as the president’s chief of staff faced mounting questions about how he dealt with the accusations.

“I think it’s fair to say that...we all could have done better over the few hours or last few days in dealing with this situation,” said White House deputy press secretary Raj Shah, referring to the administration’s handling of the allegations involving Mr. Porter.